Hamiltonian ODEs in the Wasserstein space of probability measures

In this paper we consider a Hamiltonian H on P_2(R^{2d} ), the set of probability measures with finite quadratic moments on the phase space R^{2d} = R^d × R^d , which is a metric space when endowed with the Wasserstein distance W_2. We study the initial value problem dμ_t/dt+∇·(J_dv_tμ_t ) = 0, where J_d is the canonical symplectic matrix, μ_0 is prescribed, and v_t is a tangent vector to P_2(R^{2d}) at μ_t , belonging to ∂H(μ_t ), the subdifferential of H at μ_t . Two methods for constructing solutions of the evolutive system are provided. The first one concerns only the case where μ_0 is absolutely continuous. It ensures that μ_t remains absolutely continuous and v_t = ∇H(μ_t ) is the element of minimal norm in ∂H(μt ). The second method handles any initial measure μ_0. If we further assume that H is λ-convex, proper, and lowersemicontinuous on P_2(R^{2d} ), we prove that the Hamiltonian is preserved along any solution of our evolutive system, namely H(μ_t ) = H(μ_0).